Yup, when reading down the list I got to the susb9 Phrygian, and I kind of knew that it would feature in this question!
Essentially, Phrygian can be thought of as a natural minor with a b2, hence in the same format as the rest of the chords written above you could write it
m7 (b9, 11, b13)
I tend to think of it this way, and when I got the Mark Levine book I was surprised to see it as a susb9 chord instead. In the great-american-songbook tunes the pure Phrygian doesn't pop up too often. This is because -
When that scale degree is acting as a V in a minor ii V i it's turned into a dominant b9 chord by raising it's minor 3rd, in order to better push towards the minor tonic. (note the resulting dominant chord often keeps the token Phrygian b9!)
In the common iii VI ii V I progression the Phrygian iii is generally written as just a straight minor 7
These are the more common times you see Phrygian, probably down to the commonality of these songs changing chords in 4ths. There are pieces I can think of that rise up I, ii, iii etc. But again it only really appears as a straight minor7 chord (or maybe minor11), no phyrigian extensions.
The reason being is that the b9 AND b13 extensions on a minor chord are considered 'not very flattering' and not that useful. This is subjective of course, but you see very few minor b13 or minor b9 chords used in western music (I don't know about elsewhere). I've seen the b6 called an avoid note, as well (though this is a little misleading, it more means 'be aware it will clash' rather than 'don't use it'). So the Phrygian's natural extensions, when played over a minor7 chord, don't really do it any favours. Or even if some like the sound it's certainly harder to voice it and voice lead it in a way that works as easily as the rest of the extended diatonic chords.
However, voicing it as a susb9 gets rid of this. You have all the notes needed in there to make this chord, this chord utilitises the characteristic sounds of the mode and is arguably a more practical alternative to the angular min7b9 or min7b13 chord. It's a cool sounding chord and a good way of getting a Phrygian sounding chord thats not too clashy.
That said, they list all the intervals in the above list of chords, so by the time you play your susb9 with the #9 and b13 added back in it's just as clashy as the minor Phrygian chord was. But play eg. 1, 4, b9, 4, 5 on a guitar and it sounds very very useable. The Mark Levine book is slightly contentious because it popped up at a time when relating a scale to every chord was 'the thing' in jazz and it kind of presents that info as if it's 'the only way' but thats not at all true, and not how the majority of the earlier jazz musicians approached playing, really, until the modal thing hit, and even then it was a gradual change. So it's a great resource with tonnes of info but just treat it as one persons perspective. I'm a bass player and in the Mark Levine book when he said that bass players tune to an Asusb9 chord that was news to me, none of my teachers had ever mentioned it and I'd never seen ANYONE do it in all the gigs I've ever been to :D I first encountered Phyrigian being a susb9 in that book, and I've met many astronomically good jazz musicians who are oblivious to that which Mark sets out as gospel, but I CAN see the practical function of it, and it's a unique sound.
So anyway, there are some reasons you may want to voice phyrigian as a susb9 rather than a minor chord if you play a chordal instrument, but I still think of it primarily as a minor mode when I see it come up in it's diatonic form (ie, not converted to dominant etc.)
As for the 7th mode of the harmonic minor scale, the modes of this scale are used a fair bit less than those of the major and melodic minor scales, and as such there is a bit less of a strong convention for naming them, you could arguably call it a few things but the authors intention was to explain a 'base' chord and then three easy to digest extensions for the purposes of memorization, it seems. The altered (melodic AND harmonic) is a bit of a funny one, as when you first play the mode you'll notice it, named one way, has two seconds (a flat one and a sharp one) or named another way two 3rds (a flat one and a natural one), also, the major 3rd we get is actually a flat 4th degree... oddness! Really it is 1, b2, b3, b4, b5, b6, b7/6 (melodic/harmonic) but we don't tend to think of it that way.
As the melodic minor altered scale generally functions as a dominant chord we tend to ignore the flat 4 oddness, and just say it's 1, 3 and b7 with b9, #9, b5 and #5. It's just a dominant chord with all alterations available, hence named altered.
For this one, the harmonic minor one, we can't say it functions as a dominant because it has no b7. Indeed, a major with a 6th is often a comfortable substitute for a major7 chord. So the author has tried to name the base chord in such a way that he can place the extensions on the top and create a full chord.
With the +6 he is describing the 1, 3, #5, 6, and with the added extensions you get the full scale. You're right, he could have just added the 13 on top. In that way his spelling would become
plus (b9 #9 #11 13) (I use the WORD 'plus' because the forum formatting turns a + symbol to a bullet point!)
Remember, if we are thinking of it as a major type chord then what we are really doing is thinking of the b4th degree of the scale as a major third, meaning we end up with two 2nd's, or two 9ths. This is against standard convention of 'one of each interval' but in these dusty corners the interval naming system DOES get a bit stretched, we have to just accept that that's how it is!
Arguably though, you could think of it as a minor type chord but then you have to deal with a flat 11, you could probably voice it a few other ways but I suspect the authors suggestion is one of the better, more succinct ways of expressing it in this format. I'm not too keen of thinking of the 5th note of the scale as a #11, I would still think b5th personally but really the purpose of such an exercise is just to find a way that YOU remember it, so work out his chord, change the naming a bit if needed and then go out and use it! There is far from standard convention on the chords from the harmonic minor.
As long as you understand the theory I wouldn't get too hung up on the second one, usually we can utilitise the harmonic minor in our playing by just playing the basic harmonic minor scale and remember what root notes we can place it on in order to create different effects over given chords in a piece. We don't need to worry too much about the underlying chords in the same way we do in the major scale. However, it's always fun to try and find new sounds!